Linked Questions

5 votes
1 answer
570 views

Infinite dimensional involutions: infinitely large sets of multivariate polynomials self-inverse under self-substitution

Examples of infinite dimensional involutions Edit 2/25/23, as suggested by YCOR below: (Start) The first return on a Google search on involution--from late Latin 'a rolling up'--gives the Oxford ...
0 votes
1 answer
333 views

Log associahedra and log noncrossing partitions--raising ops and symmetric function theory for $A_n$ (references)

Where do the following three sets $[LA]$, $[ILA]$, and $[LN]$ of partition polynomials appear in the literature? There are two sets of partition polynomials, not in the OEIS, that serve as the ...
1 vote
1 answer
142 views

Refinement of face vectors of the simplicial noncrossing hypertree complexes of McCammond

Einziger on page 65 of "Incidence Hopf algebras: Antipodes, forest formulas, and noncrossing partitions" presents the antipode of a noncrossing partition Hopf algebra as a graded sequence of partition ...
73 votes
9 answers
15k views

What is Lagrange Inversion good for?

I am planning an introductory combinatorics course (mixed grad-undergrad) and am trying to decide whether it is worth budgeting a day for Lagrange inversion. The reason I hesitate is that I know of ...
9 votes
1 answer
791 views

Combinatorics for the action of Virasoro / Kac–Schwarz operators: partition polynomials of free probability theory

In the background sections below, I establish the relations among characterizations of the action of Virasoro / Kac–Schwarz operators of 2D gravity models presented in terms of Laurent series by ...
95 votes
36 answers
16k views

The concept of duality

I have been thinking for sometime about asking this question, but because I did not want to have two "big-list" questions open at the same time, I did not ask this one. Now its time has come....
7 votes
0 answers
563 views

Guises of the noncrossing partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious ...
2 votes
0 answers
333 views

Combinatorics of iterated composition of noncrossing partition polynomials

A combinatorial problem arises in relating connected and disconnected Green functions associated to a "zero-dimensional" quantum field theory presented by Brezin, Itzykson, Parisi, and Zuber ...
6 votes
3 answers
2k views

An Intriguing Tapestry: Number triangles, polytopes, Grassmannians, and scattering amplitudes

What are the roles that the classic number arrays-- Eulerian, Narayana--play in the application of totally non-negative Grassmannians, or amplituhedrons, to string / twistor scattering theory? (This ...
29 votes
0 answers
3k views

Why do polytopes pop up in Lagrange inversion?

I'd be interested in hearing people's viewpoints on this. Looking for an intuitive perspective. See Wikipedia for descriptions of polytopes and the Lagrange inversion theorem/formula (LIF) for ...
29 votes
7 answers
3k views

Why are we interested in permutahedra, associahedra, cyclohedra, ...?

The following families of polytopes have received a lot of attention: permutahedra, associahedra, cyclohedra, ... My question is simple: Why? As I understand, at least the latter two were ...
11 votes
1 answer
973 views

What is a good introduction to cluster algebras from surfaces?

What is a good reference for cluster algebras from surfaces, with a view to their connection to Teichmuller theory? In my view, that means it should start off with unpunctured surfaces (and in fact,...
38 votes
17 answers
8k views

Examples of "unsuccessful" theories with afterlives

I am looking for examples of mathematical theories which were introduced with a certain goal in mind, and which failed to achieved this goal, but which nevertheless developed on their own and ...
38 votes
1 answer
4k views

Why is there a connection between enumerative geometry and nonlinear waves?

Recently I encountered in a class the fact that there is a generating function of Gromov–Witten invariants that satisfies the Korteweg–de Vries hierarchy. Let me state the fact more precisely. ...
11 votes
0 answers
604 views

Inversion, Koszul duality, combinatorics and geometry

According to this MO answer Koszul duality is related to operations on generating series; 1) multiplicative inversion for quadratic algebras, 2) compositional inversion for quadratic operads, 3) ...

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